Effective picture of bubble expansion

TL;DR

This paper develops an effective equation of motion for expanding bubbles during cosmic phase transitions, incorporating a recently calculated $ ext{γ}^2$-scaling thermal friction, and explores its impact on gravitational wave signals.

Contribution

It introduces a novel EOM framework accounting for $ ext{γ}^2$-scaling friction and proposes a phenomenological parameterization for out-of-equilibrium effects in bubble expansion.

Findings

The efficiency factor for bubble collisions is larger with $ ext{γ}^2$-scaling friction.

Gravitational wave signals are slightly enhanced due to increased collision efficiency.

The $ ext{γ}^2$-scaling friction suppresses bubble collision contributions compared to $ ext{γ}$-scaling.

Abstract

Recently the thermal friction on an expanding bubble from the cosmic first-order phase transition has been calculated to all orders of the interactions between the bubble wall and thermal plasma, leading to a $\gamma^2$-scaling instead of the previously estimated $\gamma^1$-scaling for the thermal friction exerted on a fast-moving bubble wall with a Lorentz factor $\gamma$. We propose for the first time the effective equation of motion (EOM) for an expanding bubble wall in the presence of an arbitrary $\gamma$-scaling friction to compute the efficiency factor from bubble collisions, which, in the case of $\gamma^2$-scaling friction, is found to be larger than the recently updated estimation when the bubble walls collide after starting to approach a constant terminal velocity, leading to a slightly larger signal of the gravitational waves background from bubble collisions due to its quadratic dependence on the bubble collision efficiency factor, although the $\gamma^2$-scaling friction itself has already suppressed the contribution from bubble collisions compared to that with $\gamma^1$-scaling friction. We also suggest a phenomenological parameterization for the out-of-equilibrium term in the Boltzmann equation that could reproduce the recently found $(\gamma^2-1)$-scaling of the friction term in the effective EOM of an expanding bubble wall, which merits further study in future numerical simulations of bubble expansion and collisions.